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The Slantlet Transform
Ivan W. Selesnick, Member, IEEE
Abstract— The discrete wavelet transform (DWT) is usually
carried out by filterbank iteration; however, for a fixed number
of zero moments, this does not yield a discrete-time basis that is
optimal with respect to time localization. This paper discusses the
implementation and properties of an orthogonal DWT, with two
zero moments and with improved time localization. The basis is
not based on filterbank iteration; instead, different filters are used
for each scale. For coarse scales, the support of the discrete-time
basis functions approaches two thirds that of the corresponding
functions obtained by filterbank iteration. This basis, which is
a special case of a class of bases described by Alpert, retains
the octave-band characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given,
an efficient implementation of the transform is described, and
improvement in a denoising example is shown. This basis, being
piecewise linear, is reminiscent of the slant transform, to which
it is compared.
I. INTRODUCTION
D
ISCRETE wavelet transforms (DWT’s) are useful in a
variety of applications (such as estimation, compression,
and fast algorithms), partly because they can provide relatively
efficient representations of piecewise smooth signals [9], [30].
The degree to which a wavelet basis can successfully yield
sparse representations of such signals depends on the timelocalization and smoothness properties of the basis functions.
For example, signal smoothing by the nonlinear thresholding
of DWT coefficients [14] preserves edges reasonably well—in
part, because the support of each basis function is short with
respect to its bandwidth. (Thus, we have the “constant-Q”
behavior, or octave-band characteristic, of the associated filterbank.) However, a fundamental tradeoff exists between time
localization and smoothness characteristics, and it is desirable
to obtain a good tradeoff between these two competing criteria
in designing wavelet bases. As is usual for DWT’s, in this
paper, the lengths of the discrete-time basis functions and
their moments are the vehicles by which time localization and
smoothness properties are achieved.
Although the usual filterbank iteration provides a simple
way to generate an orthogonal discrete-time basis having
an octave-band characteristic, for a fixed number of zero
moments, it does not yield a discrete-time basis that is optimal
with respect to time localization. This paper examines a special
Manuscript received February 17, 1998; revised November 9, 1998. This
work was supported by the Alexander von Humboldt Foundation. The
associate editor coordinating the review of this paper and approving it for
publication was Dr. Akram Aldroubi.
The author is with Electrical Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]).
Publisher Item Identifier S 1053-587X(99)03233-X.
case of a class of bases originally described by Alpert in
[4]–[6] in a multiwavelet context, the construction of which
relies on Gram–Schmidt orthogonalization. We describe the
basis from a filterbank viewpoint, give explicit solutions for
the filter coefficients, and describe an efficient algorithm for
the transform.
The DWT described in this paper is based on a filterbank
structure where (as in [2] and [3]) different filters are used
for each scale. Nevertheless, a very simple efficient algorithm
based on recursion is available. For the DWT filterbank
described here, the support of the discrete-time basis functions
is reduced (by a factor approaching one third for coarse
scales) while retaining the basic characteristics of the twoband iterated filterbank tree. This basis retains the octave-band
characteristic and leads cleanly to a DWT for finite length
signals (boundary issues do not arise, provided the data length
is a power of 2). The filters are piecewise linear but are
discontinuous—for coarse scales, they converge to piecewise
linear, discontinuous functions.
The basis, being piecewise linear, is reminiscent of the
slant transform to which it is compared. However, the basis
functions of the slant transform, like the Hadamard transform
for example, are nonzero over all of the domain, whereas the
basis functions described in this paper become progressively
more narrow, giving a multiresolution decomposition. Hence,
we have the name slantlet for the transform described here.
The slantlet basis appears especially well suited for treating
piecewise linear signals, as is supported by the denoising
example below.
II. SLANTLET FILTERBANK
It is useful to consider first the usual iterated DWT filterbank
and an equivalent1 form, which is shown in Fig. 1. The
“slantlet” filterbank described here is based on the second
structure, but it will be occupied by different filters that are
not products. With the extra degrees of freedom obtained by
giving up the product form, it is possible to design filters of
shorter length while satisfying orthogonality and zero moment
conditions, as will be shown.
For the two-channel case, the shortest filters for which the
filterbank is orthogonal and has zero moments are the wellzero
known filters described by Daubechies [13]. For
and
are of length 4. For this
moments, those filters
, the iterated filters in Fig. 1
system, which is designated
1 Note that interleaving the third and fourth channels of the second structure
gives the third channel of the first structure. Because that difference is
unimportant for our purpose, in this paper, the two structures will be
considered equivalent
1053–587X/99$10.00  1999 IEEE
SELESNICK: SLANTLET TRANSFORM
Fig. 1.
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Two-scale filterbank and an equivalent structure.
Fig. 3.
Three-scale filterbank and an equivalent structure.
The filters shown on the right-hand side of Fig. 2 are
D
Fig. 2. Comparison of two-scale iterated 2 filterbank (left-hand side) and
two-scale slantlet filterbank (right-hand side).
are of length 10 and 4. Without the constraint that the filters are
zero moments
products, an orthogonal filterbank with
can be obtained where the filter lengths are 8 and 4, as shown
system. That is
in Fig. 2, side by side with the iterated
a reduction by two samples, which is a difference that grows
with the number of stages, as will be shown.
Fig. 3 illustrates a three-scale filterbank tree for the DWT
and, again, an equivalent structure. The three-scale iterated
filterbank tree analyzes signals at three scales with filters
of length 4, 10, and 22, as illustrated on the left-hand side
of Fig. 4. On the other hand, the filterbank shown on the
right-hand side of Fig. 4 analyzes a signal at three scales with
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 5, MAY 1999
6) The slantlet filterbank is less frequency selective than
the traditional DWT filterbank due to the shorter length
of the filters. The time localization is improved with a
degradation of frequency selectivity.
7) The slantlet filters are piecewise linear.
It must be admitted that although both types of filterbanks
posses the same number of zero moments, the smoothness
properties of the filters are somewhat different. In Figs. 2
and 4, the slantlet filters have greater “jumps” than do the
iterated
filters—that is, they have a greater maximum
difference between adjacent sample values. The Haar basis,
with its discontinuities, is suitable for analyzing piecewise
constant functions. Likewise, the slantlet filterbank appears
appropriate for the analysis of piecewise linear functions with
discontinuities, as illustrated in the denoising example below.
The ability to model discontinuities is also relevant for other
applications, like edge detection and change point analysis,
in which the detection of abrupt changes in an otherwise
relatively smooth but unknown function is considered [24],
[27].
We also wish to mention that symmetry of the filters is an
important property in some applications, especially in image
processing. While the filters described here are not symmetric,
are paired with their time-reversed versions
the filters
so that the effect of time reversing the input signal on the
channels is merely an interchange of adjacent the channels.
A. Notation
D
Fig. 4. Comparison of three-scale iterated 2 filterbank (left-hand side) and
three-scale slantlet filterbank (right-hand side).
filters of length 4, 8, and 16. This reduction in length, while
maintaining desirable orthogonality and moment properties, is
possible because these filters are not constrained by the product
form arising in the case of iterated filterbanks.
We make several comments regarding Figs. 2 and 4.
1) Each filterbank (equivalently, discrete-time basis) is orthogonal. The filters in the synthesis filterbank are obtained by time reversal of the analysis filters.
2) Each filterbank has two zero moments. The filters (except for the lowpass ones) annihilate discrete-time polynomials of degree less than 2.
3) Each filterbank has an octave-band characteristic.
4) The scale-dilation factor is 2 for each filterbank. Between scales, the filters dilate by roughly a factor of
2. (In the slantlet filterbanks, they dilate by exactly a
factor of 2.)
5) Each filterbank provides a multiresolution decomposition. By discarding the highpass channels and passing
only the lowpass channel outputs through the synthesis
filterbank, a lower resolution version of the original
signal is obtained.
,
We will denote, by scale , the scale with which
, and
analyze a signal. The length of the filters
for scale will be proportional to . That is approximately
true for iterated filterbanks; however, it is exact for slantlet
,
, and
filterbanks. In general, the support of
will be
.
We should clarify the way in which the slantlet filterbanks
in Figs. 2 and 4 are generalized to scales. That is done as
channels. The lowpass
follows. The -scale filterbank has
. The filter adjacent to the lowpass
filter is to be called
. Both
and
are to
channel is to be called
be followed by downsampling by . The remaining
channels are filtered by
and its shifted time-reverse for
. Each is to be followed by downsampling by
. It follows that the filterbank is critically sampled.
appears
Note that in the slantlet filterbank, each filter
does not appear
together with its time reverse. While
with its time reverse, it always appears paired with the filter
. In addition, note that the -scale and
-scale
for
filterbanks have in common the filters
and their time-reversed versions.
B. Derivations
,
, and
are piecewise
That the sought filters
linear is central to the following derivation. When coupled
with the zero moments, it simplifies orthogonality conditions.
,
, and
are each linear
First, suppose that
and over the interval
over the interval
, as in Figs. 2 and 4. Suppose, in
SELESNICK: SLANTLET TRANSFORM
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addition, that
and
have two zero moments—that
is, their inner products with linear polynomial sequences are
and
annihilate “ramps”]. With
zero [as filters,
, over the support of
, the functions
,
,
are linear so that orthogonality between scales
and
is immediate. The same is true for the appropriately shifted
,
, and their time-reversed versions.
versions
is to be linear over
Because the sought-after filter
the two above-mentioned intervals, it is described by four
parameters and can be written as
for
for
Therefore, to obtain
such that the sought-after -scale
filterbank is orthogonal with two zero moments requires oband
so that we have
taining parameters
the following.
is of unit norm.
1)
2)
is orthogonal to its shifted time reverse.
3)
annihilates linear discrete time polynomials.
Each of the conditions can be written as an algebraic equation
and
to
in terms of the four parameters
obtain a multivariate polynomial system of equations. The
conditions are nonlinear in the four parameters; however, with
assistance from the computer algebra systems Maple [11] and
Singular [16] (for the computation of Gröbner bases), we
:
obtain the following expressions for
terms of eight unknown parameters
and
.
for
for
for
for
The orthogonality and moment conditions require the following.
and
are of unit norm.
1)
2)
and
are orthogonal to their shifted versions.
3)
annihilates linear discrete time polynomials.
By expressing the orthogonality and moment conditions as
a multivariate polynomial system, we obtain the following
and
:
solution for
for
for
where
Note that the parameters
and
depend on .
and
. Using, again,
The same approach works for
and
can be written in
a piecewise linear form,
In these expression for
,
, and
, the signs of any
of the square roots can be negated. Doing so merely negates
or time reverses the sequences.
and
specialize to the Daubechies
We note that
, as expected.
length-4 filters for
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C. Support Length
In Figs. 2 and 4, it was seen that the support of the slantlet
filters is less than those of the filters obtained by filterbank
iteration. It is interesting to note the difference for the general
-scale case. The iterated filterbank, with Daubechies length-4
. On the
filters, analyzes scale with a filter of length
other hand, the slantlet filterbank analyzes scale with the filter
of length
. That gives a reduction of
samples
for scale . The ratio tends to two thirds as increases (for
coarser scales). That reduction in the support of the analysis
filters is precisely what was sought.
D. Multiresolution Spaces
To clarify the multiresolution spaces generated by the filterbanks described in this paper, it is convenient to define
appropriate function spaces, as is usually done.
Span
(1)
(2)
Span
(3)
Span
(4)
(5)
(6)
(7)
(8)
Each line above corresponds to the decomposition by an
-scale filterbank, the last line being that of a four-scale
filterbank. The nesting of approximation spaces generated by
the four-scale filterbank is expressed as
E. Relationship with
reproduces the characteristics of a tree structured two-band
system.
It should also be noted that the way in which the filters
and
complete the filterbank, given the maximally
regular lowpass branch, differs from that suggested in [29].
In [29], a completion of the filterbank that approximates a
equal
uniform division of the frequency spectrum into
-band
bands is suggested. Indeed, the motivation for the
system in [29] is not the improvement of time-localization
properties while preserving the essential time–frequency tiling
of the two-band DWT, but it is to provide a different tiling
of the time–frequency plane that might better suite certain
applications.
Certainly, given an -band lowpass filter, we can complete
the filterbank in a number of ways, one being the approximate
uniform division of frequency and a second being a division
in frequency similar to that provided by an iterated two-band
system. In this paper, we have chosen the second and have
used the greater generality to improve the time localization of
the resulting basis.
F. Finite-Length Signals
The orthogonal discrete wavelet transform based on filterbank iteration is usually adapted to finite (power of 2) length
data by periodizing the signal. Each output of the analysis
filterbank is then periodic, and in this manner, an orthogonal
transformation can be constructed for a finite interval. The
same can be done for the slantlet filterbank, with results that
are especially clean, due to the lengths of the the filters being
powers of 2. Consider the orthogonal matrix of dimension
representing the transform associated with an -scale filterbank.
.
In Fig. 5, a 16 16 example is illustrated for
, is
The first row of the matrix, which corresponds to
simply a constant. The effect of periodizing the input results
in an overlapping effect for
-band Wavelet Bases
It should be noted that a relationship exists between the
bases described in this paper and those described in [17]–[19],
[29]. Those references describes a generalization of the twoband Daubechies wavelet basis to -bands or, equivalently,
an -channel orthogonal filterbank with zero moments for
and . In particular, for an -band system, these
general
references describe the shortest lowpass (scaling) filter with a
of zero moments. We wish to note that
specified number
, this maximally regular filter (in the terminology of
for
described above
[29]) is identical to the lowpass filter
).
(with
As noted in [29], given the lowpass branch of an orthogonal
filterbank, the remaining filters are not uniquely determined,
in contrast with the two-band case. This makes the design of
remaining channels more difficult, and although methods for
filters to complete the filterbank are
obtaining a set of
described in [29] (see also [31]), it can be difficult to control
the characteristics of the resulting filters and, in particular,
, the filters
and
to regulate their lengths. For
described above give a way to complete the filterbank,
given the maximally regular lowpass branch, in a way that
(9)
(10)
. The second row, corresponding
for
, is a linear function, as is shown in Fig. 5. Periodizato
, as it does for
tion results in an overlapping effect for
(11)
(12)
. Each of the remaining rows
for
, its time reverse,
of the matrix consists of the sequences
, for
. Except for the
and their shifts by
first two rows, there is no overlapping effect at the boundaries
of the matrix, as the supports are powers of two.
SELESNICK: SLANTLET TRANSFORM
Fig. 5.
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N = 16) basis. Vectors of the 16 2 16 orthogonal matrix associated with the four-scale slantlet filterbank.
Slantlet (
G. Comparison with Slant Transform
Interestingly, the Haar basis can be obtained by downsampling the Walsh basis. Both are piecewise constant, but the
Walsh transform serves as a minimal complexity DCT for
frequency analysis, whereas the Haar transform, having basis
functions of progressively shorter widths, gives a multiresolution decomposition. A piecewise linear basis that follows
the spirit of the Walsh transform (in performing frequency
analysis) is the slant transform [1], [7], [15], [23], [25], [26],
[33], which has been used in Intel’s “Indeo” video compression
algorithm [8]. In a loose sense, the transform described in this
paper is to the slant transform what the Haar transform is to
the Walsh transform. The analogy is only loose, however, and
their similarities suggest the name slantlet transform for the
transform described in this paper.
to compute
. Writing the output sample of channel
an inner product
as
(13)
(14)
H. Efficient Implementation
A key to the efficient implementation of the usual DWT
is its tree structure. The long filters used to analyze coarse
scales are implemented by a sequence of convolutions and
downsampling. In the slantlet filterbank, it appears initially that
an efficient implementation is not available for the lack of tree
structure. However, an efficient implementation is possible, as
is shown here. The efficient algorithm given below resembles
closely the iterative procedure used to implement an iterated
filterbank tree; therefore, the computational complexity is of
the same order, although the code complexity may increase.
Because the filters are piecewise linear, each filter can be
represented as the sum of a DC and a linear term. Due to the
, only four terms are needed
simple form of the filters
(15)
(16)
where
(17)
(18)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 5, MAY 1999
are types of DC and linear moments at scale of the input
. The same expressions are valid for the projections
signal
onto the time-reversed versions of
, but the
of
and
are to be modified.
constants
Note that as in the Haar DWT, the moments can be
computed efficiently by a simple recursive algorithm that starts
. The DC and linear moments at scale
with the fine scale
can be computed from the DC and linear moments at the
by
next finer scale
(19)
(20)
The inverse can also be computed efficiently by making
and
. For the efficient
use of the values
and
computation of the inverse, we first compute
from the DC and linear slantlet coefficients; we then
and
for decreasing values of by
compute
and
using the slantlet coefficients. Finally, with
updating
, the original signal
is obtained from
and
via
the relation
(21)
(22)
Therefore, even though the design of the filterbank is not
based on an iterated filterbank, the computation of its output
can be made efficient by a recursive method.
J. Denoising Example
In this denoising example, the behavior of the slantlet basis
is compared with other wavelet bases having two vanishing
moments: the
basis, the biorthogonal-2,2 and -2,4 bases
(see [13, p. 273]), and the piecewise linear semi-orthogonal
(spline) bases (see [32, p. 147]). For the nonorthogonal bases,
the DWT was carried out with symmetric extensions. A hard
threshold was applied uniformly to each scale. We chose the
signal to be the “Houston skyline” function by Guo because
it is piecewise linear and has numerous discontinuities. Fig. 6
illustrates the results. Denoising with the slantlet transform
yields the same artifacts and noise spikes, but for this example, they were generally reduced. Varying the threshold
used and averaging over 200 realizations for each threshold,
the curve illustrating the root-mean-square error in Fig. 6
was obtained. That figure shows that for this example, the
slantlet transform gives an improvement. It indicates that on
average, for thresholds between 0.12 and 0.24, the error with
for any
the slantlet is smaller than that obtained with
threshold. That a wider choice of thresholds gives such results
is important because in practice, of course, the best threshold
for a particular example is unknown. It is interesting to note
that for thresholds above 0.24, the semi-orthogonal (spline)
bases perform better than do the biorthogonal bases in this
example. Certainly, the most appropriate basis depends on the
data and the noise level.
It should be noted that there are a variety of denoising techniques that go beyond simple thresholding that can produce
dramatically improved results. For example, we have the shiftinvariant transform of [10] and [20] mentioned above and the
hidden-Markov model-based approach of [12].
I. Shift Variance and Redundant Transform
It should be noted that slantlet filterbanks are more time
varying than those based on filterbank iteration. Consider the
highpass branch of the tree-structured filterbank in Figs. 1 and
3. The highpass channel is periodically time varying due to
the downsampling, with a period of 2. On the other hand, the
highpass channels of the slantlet filterbank are periodically
time varying with period 4.
It is seen that the improvement with regard to time localization costs us not only the simple tree structure but also
costs one greater shift variance. In some applications, that is a
disadvantage. However, in denoising, the loss of shift variance
can be overcome by turning to a redundant transform. With
such a transform, shift invariance is retrieved by effectively
including all shifts of the data and comes at the expense of
a redundant representation. Interestingly, it has been shown
that denoising via wavelet thresholding can yield superior
results when carried out with this shift-invariant redundant (or
stationary) wavelet transform [10], [20]. If memory and runtime requirements permit, the use of a redundant transform
for denoising can be advantageous. In this case, redundant
denoising is an application where the greater shift variance
of the slantlet filterbank is not expected to be a significant
drawback. A redundant shift-invariant version of the slantlet
transform, in the sense of [10], is straightforward to derive and
can be implemented in a similar way.
K. Underlying Continuous-Time Wavelets
As noted in the Introduction, the slantlet basis is a special
case of the multiwavelet bases described by Alpert [4]–[6],
comprised of scaling functions and wavelet functions with
vanishing moments. The continuous-time multiwavelet basis
is piecewise linear and discontinuous.2
of [6] with
However, it is important to note that for these bases, the relationship between continuous-time and discrete-time versions
is not as simple as it is for scalar wavelet bases (wavelet
bases based on a single scaling function). In the scalar case,
the discrete-time basis is obtained by iterated filtering and
upsampling. However, the filterbank associated with Alpert’s
does not yield
continuous-time multiwavelet bases with
the discrete-time slantlet basis due to important differences
between scalar- and multiwavelet bases, as highlighted in [28];
in the terminology of [21] and [22], the multiwavelet basis is
not balanced. To obtain a discrete-time version of the basis,
Alpert used a Gram–Schmidt orthogonalization and considers
the general case of vanishing moments, whereas we use
Gröbner bases to derive explicit solutions for the special case
of 2 vanishing moments. The explicit solutions for the filters at
0
2 Alpert’s basis has
0 (t) symmetric: 0 (t) = 0 (t T ), and 1 (t) antisymmetric: 1 (t) =
T ). It is immediate that pairwise symmetry
1 (t
can be obtained by taking their sum and difference.
0
0
SELESNICK: SLANTLET TRANSFORM
Fig. 6.
1311
Denoising via hard thresholding with the iterated
each scale are useful because they are required for the efficient
implementation of the transform described in Section II-H.
The approach taken by Alpert addresses the extension
of these bases to a higher number of vanishing moments;
however, higher order extensions entail a higher number of
separate functions/filters to analyze a signal at a single stage.
The number of filters increases according to the order. For
,
bases that are piecewise polynomial with degree
vanishing moments requires wavelet functions/filters.
L. Multidimensional Filterbanks
The generalization to the multidimensional case is not as
straightforward as it is for iterated filterbank trees. Neverthe-
D2 filterbank and the slantlet DWT.
less, a 2-D filterbank can be obtained in a similar manner
that is composed of separable filters, although the filterbank
itself is not separable. In the 2-D case, the area of support
of the filters approaches
that of the iterated 2system. That is a reduction of the area of support by
D
over one half, suggesting that in more than one dimension,
the tradeoff between zero moments and time- localization
becomes more significant. However, the smoothness properties
transforms are
of the 2-D slantlet and the separable 2-D
different as well. While the highpass and bandpass filters of
the 2-D slantlet filterbank annihilate linear polynomials in two
; the separable 2-D
filterbank
variables,
.
annihilates polynomials of the form
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TABLE I
COMPARING THE ITERATED 2 AND SLANTLET FILTERBANKS
D
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It should be emphasized that the slantlet transform is most
appropriate for data that is piecewise linear and cannot be
expected to be useful in the compression of natural images,
for example. A description of the details of the 2-D slantlet
transform will be available from the author.
III. CONCLUSION
The smoothing of data while preserving edges relatively
well is an essential advantage of wavelets in denoising, and it
depends in part on both the short support of the basis functions
with respect to their scale and their number of vanishing
moments. In addition, in the application of wavelet bases to
image compression, the time localization and the number of
zero moments of the basis are both important. Good timelocalization properties lead to good representation of edges.
Approximation order is important for sparse representation
(compression) of smooth regions. However, short support and
zero moments are competing criteria in the construction of
wavelet filterbanks.
In this light, this paper presents an orthogonal filterbank for
the discrete wavelet transform with two zero moments, where
the filters are of shorter support than those of the iterated
filterbank tree. Although not based on an iterated filterbank
tree, the filterbank described in this paper retains the main
desirable characteristics of the usual DWT filterbank, namely,
orthogonality, an octave-band characteristic, a scale-dilation
factor of 2, and an efficient implementation. Table I summarizes a comparison. A transform for finite length signals based
on this filterbank is particularly clean due to the filter lengths
being exact powers of two. The basis appears particularly well
suited for piecewise linear signals, as does the Haar basis
for piecewise constant signals. Improvement in a denoising
example was also shown.
Matlab programs for the slantlet transform, its inverse,
a shift-invariant (redundant) variant, and a 2-D version
are available from the author or via the Internet at
http://taco.poly.edu/selesi/.
ACKNOWLEDGMENT
The author wishes to thank H. W. Schüßler and P. Steffen
of the Lehrstuhl für Nachrichtentechnik, Universität ErlangenNürnberg, H. Guo for providing his “Houston Skyline” function, and the anonymous reviewers.
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SELESNICK: SLANTLET TRANSFORM
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Ivan W. Selesnick (M’98) received the B.S.,
M.E.E., and Ph.D. degrees in electrical engineering
in 1990, 1991, and 1996, respectively, from Rice
University, Houston, TX.
He received a DARPA-NDSEG fellowship in
1991. He has been employed by McDonnell
Douglas and IBM, working on neural networks
and expert systems. He spent part of 1997 at the
the Lehrstuhl für Nachrichtentechnik Universität
Erlangen-Nürnberg, Erlangen, Germany. He is
currently an Assistant Professor in the Electrical
Engineering Department, Polytechnic University, Brooklyn, NY. His current
research interests are in the area of digital signal processing, in particular,
fast algorithms for digital signal processing, digital filter design, the theory
and application of wavelets, and the application of Gröbner bases.
Dr. Selesnick’s Ph.D. dissertation received the Budd Award for Best
Engineering Thesis at Rice University in 1996 and an award from the RiceTMC chapter of Sigma Xi. In 1997, he received an Alexander von Humboldt
Award. He is a member of Eta Kappa Nu, Phi Beta Kappa, Tau Beta Phi,
and Sigma Xi.